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Applicability of deterministic global optimization to the short-term hydrothermal coordination problem

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(1)Programa de Doctorat: Departament: Director i tutor:. Aplicacions Tècniques i Informàtiques de l’Estadı́stica, la Investigació Operativa i l’Optimització Estadı́stica i Investigació Operativa Dr. Narcı́s NABONA FRANCISCO. Applicability of Deterministic Global Optimization to the Short-Term Hydrothermal Coordination Problem. TESI DOCTORAL presentada per Albert FERRER BIOSCA a LA UNIVERSITAT POLITÈCNICA DE CATALUNYA per a optar al grau de DOCTOR EN MATEMÀTIQUES. BARCELONA, GENER 2004.

(2) Albert Ferrer Biosca Departament de Matemàtica Aplicada I Universitat Politècnica de Catalunya Avgda. Dr. Gregorio Marañón, 42-50 08028-Barcelona (Spain) alberto.ferrer@upc.es.

(3) A la Matilde i al meu pare Lluis i a la meva mare Salut i al meu germà Marcel.

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(5) Acknowledgements I am indebted to my PhD supervisor, Professor Narcı́s Nabona Francisco, who has provided indispensable encouragement, help and support during the time-consuming process of preparing this Thesis. I would like to express my deepest gratitude to Professor Juan Enrique Martı́nez Legaz for his advice and kind attention throughout the course of this work. Special thanks are due to the Professors Dinh The Luc and Alex Rubinov for their valuable comments and suggestions that have helped to improve the contents of the Thesis. I wish to express my sincere thanks to all those other persons who have contributed directly or indirectly to the process of writing this work. Last but not least, warm thanks go to my wife Matilde, my daughter Alba and my son Damià for their patience and understanding..

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(7) Contents. Notations. xi. Preliminaries. xv. Introduction. 1. 1 Overview and scope of global optimization. 9. 1.1. Historical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9. 1.2. Classification of global optimization methods . . . . . . . . . . . . .. 12. 1.2.1. Heuristic methods . . . . . . . . . . . . . . . . . . . . . . . .. 12. 1.2.2. Approximation methods . . . . . . . . . . . . . . . . . . . . .. 12. 1.2.3. Systematic methods . . . . . . . . . . . . . . . . . . . . . . .. 13. 1.3. General complementary convex mathematical structure . . . . . . .. 15. 1.4. Classes of nonconvex problems with a d.c. structure . . . . . . . . .. 18. 1.4.1. Concave minimization . . . . . . . . . . . . . . . . . . . . . .. 18. 1.4.2. Reverse convex programming . . . . . . . . . . . . . . . . . .. 18. 1.4.3. D.c. programming . . . . . . . . . . . . . . . . . . . . . . . .. 19. 1.4.4. Continuous optimization . . . . . . . . . . . . . . . . . . . . .. 20. i.

(8) ii. CONTENTS. 2 The Generation Problem and its complementary convex structure 21 2.1. 2.2. 2.3. 2.4. The short-term hydrothermal coordination of electricity generation problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 21. 2.1.1. The power hydrogeneration function in a reservoir . . . . . .. 22. 2.1.2. The Generation Problem. . . . . . . . . . . . . . . . . . . . .. 25. The Generation Problem as a d.c. program . . . . . . . . . . . . . .. 25. 2.2.1. D.c. representation of the nonlinear constraints . . . . . . . .. 26. 2.2.2. D.c. representation of the objective function . . . . . . . . . .. 27. 2.2.3. The d.c. program of reduced size . . . . . . . . . . . . . . . .. 27. Equivalent reverse convex programs . . . . . . . . . . . . . . . . . .. 28. 2.3.1. The equivalent canonical d.c. program . . . . . . . . . . . . .. 28. 2.3.2. A more advantageous equivalent reverse convex program . . .. 30. Characteristics of the hydrogeneration systems . . . . . . . . . . . .. 31. 3 How to obtain a d.c. representation of a polynomial. 35. 3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 35. 3.2. Some relevant properties of polynomials and convex functions . . . .. 35. 3.3. D.c. representation of a polynomial as a difference of convex polynomials 38 3.3.1. Alternative procedure to obtain a d.c. representation . . . . .. 41. 3.4. Advantages of using ith powers of homogeneous polynomials of degree 1 42. 3.5. D.c. representation of the power hydrogeneration function . . . . . . 3.5.1. 3.5.2. 43. D.c. representation of the homogeneous polynomial component of degree 2 . . . . . . . . . . . . . . . . . . . . . . . . .. 44. D.c. representation of the homogeneous polynomial component of degree 3 . . . . . . . . . . . . . . . . . . . . . . . . .. 46.

(9) iii. CONTENTS. 3.5.3. D.c. representation of the homogeneous polynomial component of degree 4 . . . . . . . . . . . . . . . . . . . . . . . . .. 4 The global optimization algorithm. 46 47. 4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 47. 4.2. Global optimal and global ǫ-optimal solutions . . . . . . . . . . . . .. 48. 4.3. Subdivision processes . . . . . . . . . . . . . . . . . . . . . . . . . . .. 51. 4.3.1. Simplicial subdivision processes . . . . . . . . . . . . . . . . .. 51. 4.3.2. Prismatical subdivision processes . . . . . . . . . . . . . . . .. 54. 4.4. Outline of the method . . . . . . . . . . . . . . . . . . . . . . . . . .. 57. 4.5. Initialization of the algorithm . . . . . . . . . . . . . . . . . . . . . .. 59. 4.6. Outer approximation process . . . . . . . . . . . . . . . . . . . . . .. 60. 4.7. The algorithm and its convergence . . . . . . . . . . . . . . . . . . .. 63. 4.7.1. Convergence of the algorithm . . . . . . . . . . . . . . . . . .. 65. 4.8. Appropriate linear program routine by using barycentric coordinates. 67. 4.9. Differences between the algorithm put forward and other existing algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 69. 5 How to improve a d.c. representation of a polynomial. 71. 5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 71. 5.2. The normed space of the polynomials. . . . . . . . . . . . . . . . . .. 72. 5.3. The Minimal Norm problem . . . . . . . . . . . . . . . . . . . . . . .. 73. 5.4. Relationships between semidefinite and semi-infinite programming .. 76. 5.5. The equivalent semi-infinite and semidefinite programs for solving the Minimal Norm problem . . . . . . . . . . . . . . . . . . . . . . . . .. 78. Example to compare the semi-infinite and semidefinite procedures .. 79. 5.6.

(10) iv. CONTENTS. 6 Solving semi-infinite quadratic programs. Numerical results. 89. 6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 89. 6.2. Build-up and build-down strategies . . . . . . . . . . . . . . . . . . .. 90. 6.3. Some relevant properties of linear varieties, projectors and convex programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 93. 6.3.1. Distance between a point and a linear variety . . . . . . . . .. 93. 6.3.2. Projectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 95. 6.3.3. General properties of dual programming problems . . . . . .. 97. 6.4. Analysis of the discretized problems . . . . . . . . . . . . . . . . . .. 98. 6.5. Distance from a noncentered point to the central path . . . . . . . . 102. 6.6. Relationship between the solutions to the semi-infinite and the discretized problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103. 6.7. The effect of adding a constraint . . . . . . . . . . . . . . . . . . . . 105. 6.8. The effect of deleting a constraint . . . . . . . . . . . . . . . . . . . . 116. 6.9. The build-up and down quadratic semi-infinite logarithmic barrier algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119. 6.10 Semi-infinite quadratic test problem and numerical results . . . . . . 122 6.11 Graphical description and numerical results of the instance c003d2 . 123 6.12 Optimal d.c. representation of the power hydrogeneration functions . 124. 7 Numerical results and conclusions. 131. 7.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131. 7.2. Global optimization test problems and numerical results . . . . . . . 132 7.2.1. The class of test problems HP T nXmY . . . . . . . . . . . . 132. 7.2.2. The class of test problems T nXrY . . . . . . . . . . . . . . . 133.

(11) v. CONTENTS. 7.2.3. The instance HP Br1 . . . . . . . . . . . . . . . . . . . . . . 136. 7.2.4. The instance COSr0 . . . . . . . . . . . . . . . . . . . . . . . 137. 7.3. Characteristics of the generation systems and numerical results . . . 137. 7.4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140. 7.5. Publications originated by this Thesis . . . . . . . . . . . . . . . . . 142. 7.6. Topics for further research . . . . . . . . . . . . . . . . . . . . . . . . 143. Appendix A. 145. Appendix B. 149. Bibliography. 153.

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(13) List of Tables 2.1. Characteristics of the hydrogeneration systems . . . . . . . . . . . .. 31. 2.2. The technological coefficients at each reservoir . . . . . . . . . . . . .. 33. 2.3. Bounds on the volumes and the water discharges, and the volumes stored at the beginning and at the end of the time period at each reservoir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 33. Forecasts for electricity consumption and the natural water inflow at each time interval for the reservoirs of the instances of the hydrogeneration systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 34. 3.1. Change of basis in the space of the polynomials . . . . . . . . . . . .. 45. 6.1. Dynamic mesh procedure for SIQP. 92. 6.2. Center and add constrains procedure for SIQP . . . . . . . . . . . . 110. 6.3. Delete constraints procedure for SIQP . . . . . . . . . . . . . . . . . 120. 6.4. Logarithmic barrier algorithm for quadratic semi-infinite programs . 121. 6.5. Characteristics and exact instance solutions of the semi-infinite quadratic test problem . . . . . . . . . . . . . . . . . . . . . . . . . . 123. 6.6. Instance solutions of the semi-infinite quadratic test problem by using semi-infinite algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 124. 6.7. Initial d.c. representation of the power hydrogeneration function for the reservoirs R1 and R3 . . . . . . . . . . . . . . . . . . . . . . . . 126. 2.4. vii. . . . . . . . . . . . . . . . . . ..

(14) viii. LIST OF TABLES. 6.8. Initial d.c. representation of the power hydrogeneration function for the reservoirs R2 and R4 . . . . . . . . . . . . . . . . . . . . . . . . 127. 6.9. Optimal d.c. representation of the power hydrogeneration function for the reservoirs R1 and R3 . . . . . . . . . . . . . . . . . . . . . . . . 128. 6.10 Optimal d.c. representation of the power hydrogeneration function for the reservoirs R2 and R4 . . . . . . . . . . . . . . . . . . . . . . . . 129 7.1. Parameters for the test problem HP T nXmY . . . . . . . . . . . . . 133. 7.2. Computational results for the instance HP T n2m10 . . . . . . . . . . 134. 7.3. Computational results for the instance T n2r4 . . . . . . . . . . . . . 135. 7.4. Computational results for the instance HP Br1 . . . . . . . . . . . . 136. 7.5. Computational results for the instance COSr0 . . . . . . . . . . . . 137. 7.6. Results and CP U requirements of the instances of the Generation Problem solved . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140.

(15) List of Figures 1.1. A multiextremal instance . . . . . . . . . . . . . . . . . . . . . . . .. 11. 2.1. Four intervals and two reservoirs replicated hydronetwork . . . . . .. 22. 2.2. Cross-section of a reservoir. . . . . . . . . . . . . . . . . . . . . . . .. 23. 2.3. The two basic models for obtaining the replicated hydronetwork of the instances of the hydrogeneration systems. . . . . . . . . . . . . .. 31. Feasible set of a reverse convex program as required by the modified algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 48. The sets of the ǫ-global minimizers and global minimizers of a reverse convex program as used by the modified algorithm . . . . . . . . . .. 51. 4.1. 4.2. 4.3. Exhaustive and nonexhaustive filters in a simplicial subdivision process 53. 4.4. Initialization of the modified algorithm . . . . . . . . . . . . . . . . .. 60. 4.5. The prismatic subdivision process of a prism in subprisms. . . . . . .. 61. 4.6. ǫ (x) . . . . . . . The piece-wise linear and proper convex function ψM. 62. 5.1. Least deviation decomposition (LDD) of the polynomial (x − y)2 . .. 87. 6.1. Distance between a point and a linear variety . . . . . . . . . . . . .. 94. 6.2. Description of a semi-infinite quadratic instance by using semi-infinite algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 ix.

(16) x. LIST OF FIGURES. 7.1. Plot of the objective function for the instance HP T n2m10 . . . . . . 134. 7.2. Plot of the objective function for the instance T n2r4 . . . . . . . . . 135. 7.3. Incumbent and subdivisions when a few iterations have been performed by using the modified algorithm . . . . . . . . . . . . . . . . 138. 7.4. Incumbent and subdivisions when the modified algorithm finds a global ǫ-solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138. 7.5. Incumbent and subdivisions when the modified algorithm cannot further improve the incumbent. . . . . . . . . . . . . . . . . . . . . . . . 139.

(17) Notations Generation Problem. Ne Nt dij vji li wji hij sjw s̄jw sj s̃j svb svl svq svc sdb sdl. number of reservoirs number of time intervals water discharges from reservoir j over the ith time interval volume stored in reservoir j at the end of the ith time interval forecast for electricity consumption over the ith time interval natural water inflow into reservoir j over the ith time interval power hydrogeneration function of the j th reservoir into the ith time interval headwater elevation of the j th reservoir tailwater elevation of the j th reservoir head between the headwater elevation and the tailwater elevation of the j th reservoir; sj = sjw − s̄jw equivalent head of the j th reservoir basic coefficient in the relationship between the headwater elevation and the volume stored linear coefficient in the relationship between the headwater elevation and the volume stored quadratic coefficient in the relationship between the headwater elevation and the volume stored cubic coefficient in the relationship between the headwater elevation and the volume stored basic coefficient in the relationship between the tailwater elevation and the water discharge linear coefficient in the relationship between the tailwater elevation and the water discharge xi.

(18) xii. NOTATIONS. sdq. quadratic coefficient in the relationship between the tailwater elevation and the water discharge g acceleration due to gravity dEp potential energy increment (or differential) dvj water stored increment (or differential) i t ith time interval ρij efficiency coefficient of the power hydrogeneration function hji efficiency and unit conversion coefficient; kji = ρij g/ti kji i i d.c. representation of the power hydrogeneration function hij fj − gj ϕi1 (z), ϕi2 (z) convex functions to obtain the d.c. constraints and the d.c. objective function (ϕ1 (z) − ϕ2 (z)) of the d.c. program of reduced size Vector spaces and polynomials Lm. i=0 Fi. direct sum of the vector spaces Fi he1 . . . er i the span of the set {e1 . . . er } IRm [x1 , ..., xn ] the vector space of polynomials of degree until m Hi [x1 , ..., xn ] the vector space of homogeneous polynomials of degree i IRm [X] Rm [X] = Rm [x1 , ..., xn ] where X = {x1 , ..., xn } Hi [X] Hi [X] = Hi [x1 , ..., xn ] where X = {x1 , ..., xn } P + − H , H closed half spaces defined by H = { ni=1 ai · xi = 0} P P H + = { ni=1 ai · xi ≥ 0} , H − = { ni=1 ai · xi ≤ 0} q(x1 , . . . , xn ) polynomial in the variables x1 , . . . , xn q(x0 , x1 , . . . , xn ) homogenization of the polynomial q(x1 , . . . , xn ) defined by x1 xn q(x0 , x1 , . . . , xn ) = xm 0 q( x0 , . . . , x0 ) Bk the usual bases of the monomials in Hk [x1 , ..., xn ] Bm the usual bases of the monomials in IRm [x1 , ..., xn ] ; B m = ∪m k=0 Bk Kk (C) the nonempty closed convex cone of the polynomials on Hk [x1 , ..., xn ] which are convex on the closed convex set C m K (C) the nonempty closed convex cone of the polynomials on IRm [x1 , ..., xn ] which are convex on the closed convex set C K(C) to refer to the cones Kk (C) and K m (C) indistinctly z + K(C) the cone with apex z kz(x)kp the p-norm in IR [x1 , ..., xn ] for all p = 0, 1, 2, . . . and p = ∞ kz(x)k(p,k) sometimes, it is used to indicate the p-norm in Hk [x1 , ..., xn ].

(19) NOTATIONS. xiii. Sets, functions and matrices ∅ IN IR IR IRn IRn×m B(x, δ) cone(A) int C cl C D\C ext C ∂C projIRn D f ◦g. empty set the set of the nonnegative integers (0 ∈ IN ) the set of the real numbers the set of the extended real numbers; IR := IR ∪ {−∞, +∞} IRn := IR × IR × . . . × IR, n-times IRn×m := IRn × IRm ball with center x and radius δ > 0; B(x, δ) := {z : kz − xk < δ} the cone generated by A (the smallest convex cone which contains A) interior of C; the largest open set contained in C closure of C; the smallest closed set containing C difference of the sets D and C; D \ C := {x : x ∈ D, x 6∈ C} exterior of C; ext C := int (IRn \ C) the boundary of the set C; ∂C := cl C ∩ cl (IRn \ C) the projection of D on IRn the composition of mappings g : X → Y and f : Y → Z, i.e, (f ◦ g)(x) := f (g(x)) argminf (S) the set of all global minimizers of the function f on the set S ∇f (x) gradient of the function f at the point x; ∂f (x) subdiferential of the function f at the point x; DC(A) the class of d.c. functions on the set A ⊂ IRn C 2 (A) the class of functions whose second partial derivatives are continuous everywhere on A, A open set IRn∗m the vector space of matrices with n rows and m columns Det (M ) determinant of the square matrix M ∈ IRn∗n M N when M − N is positive semidefinite for any two symmetric matrices M and N in IRn∗n.

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(21) Preliminaries Convex sets A set A ⊂ IRn is called a convex set if it contains any line segment between a pair of its points. The dimension of a convex set is the dimension of its affine hull (the smallest affine set containing A). A convex set A ⊂ IRn is said to be of full dimension if dim(A) = n. Let A, B be convex sets of IRn and let t ∈ IR. Then, the sets • A ∩ B, • A + B := {z ∈ IRn : z = x + y, x ∈ A, y ∈ B}, • tA := {z ∈ IRn : z = tx, x ∈ A, t ∈ IR}, are convex sets. Given a set X ⊂ IRn , the intersection of all convex sets which contain X is called the convex hull of X and it is denoted by conv(X). The convex P hull of X coincides with all the convex combinations of its elements ( m i=1 ti xi with Pm xi ∈ X, ti ≥ 0, i=1 ti = 1, m ∈ IN ) and it is the smallest convex set containing X. Let C ⊂ IRn be a nonempty closed convex set, and consider y 6∈ C. Then there  exists a hyperplane H := x ∈ IRn : ct x = b , with b ∈ IR and non zero c ∈ IRn such that 1. y 6∈ H + (y ∈ int(H − )), 2. C ⊂ H + , . . where H + := x ∈ IRn : ct x ≥ b and H − := x ∈ IRn : ct x ≤ b are the closed halfspaces defined by H. An immediate consequence of this property is that a xv.

(22) xvi. PRELIMINARIES. nonempty closed convex set C is the intersection of all closed halfspaces containing C. In order to express C as the intersection of halfspaces, we only need hyperplanes H which contain a boundary point of C (supporting hyperplanes of C), i.e., H ∩C 6= ∅ and C ⊂ H + or C ⊂ H − . A set M ⊂ IRn is called a cone if tM ⊂ M for all t > 0. Consider a ∈ IR then, C := a + M is called a cone with apex a. A cone which contains no line is said to be pointed in this case, 0 and a are called the vertex of M and C, respectively. A set M ⊂ IRn is a convex cone if and only if 1. tM ⊂ M for all t > 0, 2. M + M ⊂ M , which is equivalent to saying that the cone M contains all the positive linear combinations of its elements. Let A be a convex set of IRn . Denote by cone(A) the smallest convex cone that contains A, which is said to be the cone generated by A. We can see that cone(A) = ∪t>0 (tA).. Convex functions Let IR := IR ∪ {−∞, +∞} be the set of the extended real numbers with the wellknown rules of calculus with these new elements +∞ and −∞, and the meaningless situations such that +∞−∞ and ±∞/±∞, among others, that must be avoided. A function f : A → IR on a set A ⊂ IRn , is said to be an extended real-valued function on A. The sets • dom(f ) := {x ∈ A : f (x) < +∞} and • epi(f ) := {(x, t) ∈ A × IR : f (x) ≤ t} ⊂ IRn × IR are named the effective domain and the epigraph of f , respectively. If dom(f ) 6= ∅ and f (x) > −∞ for all x ∈ A, then f is said to be a proper function. A function f : A ⊂ IRn → IR is said to be convex on A when epi(f ) is a convex set in IRn × IR. This is equivalent to say that A is a convex set in IRn and f (λx1 + (1 − λ)x2 ) ≤ λf (x1 ) + (1 − λ)f (x2 ).

(23) xvii. PRELIMINARIES. for any x1 , x2 ∈ A, 0 ≤ λ ≤ 1 and the right hand side is defined. If the inequality is strict for x1 6= x2 and 0 < λ < 1, then the function is said to be strictly convex. It can be proved that if f is convex on A then f. m X i=1. λi x i. !. ≤. m X. λi f (xi ),. i=1. P. n where m ∈ IN , xi ∈ A, 0 ≤ λi ≤ 1, i = 1, . . . , m and m i=1 λi = 1. Let A ⊂ IR be a convex set. A function f : A → IR is said to be concave (strictly concave) on A when the function −f is convex (strictly convex) on A. Many properties of convex functions can be deduced from corresponding properties of convex sets.. The following algebraic properties are a direct consequence of the definition of a convex function. Let fi : A → IR, i = 1, . . . , m be proper convex functions on the convex set A ⊂ IRn , then •. Pm. i=1 αi fi (x),. αi ≥ 0, i = 1, . . . , m is convex,. • max {fi (x), i = 1, . . . , m} is convex, • sup {f (x), f ∈ F}, where F is a family of proper convex functions on A, is also a convex function. Convex functions have interesting continuity and differentiability properties which are very useful in optimization. Let A ⊂ IRn be a nonempty convex set of full dimension n. • A convex function f : A → IR is continuous at every interior point of A. If A = IRn , then f is continuous everywhere. For A 6= IRn discontinuities can only be found at the boundary of A. • If f : A → IR is differentiable on A, A open convex set, then f is convex if and only if f (y) ≥ f (x) + (y − x)t ∇f (x) for every x, y ∈ A. It is strictly convex if and only if the inequality is strict for x 6= y. • If f : A → IR is twice-differentiable on A, A open convex set, then f is convex if and only if its Hessian matrix H(x) is positive semidefinite for every x ∈ A, i.e., y t H(x)y ≥ 0 for every x ∈ A and y ∈ IRn . If H(x) is positive definite for every x ∈ A, then f is strictly convex. Given a proper function f : A ⊂ IRn → IR. A vector p ∈ IRn is said to be a subgradient of f at a point x ∈ A if f (y) ≥ f (x) + pt (y − x) for all y ∈ A. The set.

(24) xviii. PRELIMINARIES. of all subgradients at the point x is said to be the subdifferential of the function f at the point x, and it is denoted by ∂f (x). A function f is called subdifferentiable at x if ∂f (x) 6= ∅. Let f : A ⊂ IRn → IR be a proper convex function on A, A a convex set. For any bounded set S ⊂ int(dom(f )) the set S. ∂f (x). x∈S is nonempty and bounded. In particular, ∂f (x) is nonempty and bounded at every x ∈ int(dom(f )). If f is differentiable at x ∈ A then ∂f (x) = {∇f (x)}.. D.c. functions Let f be a real valued function defined on a convex set A ⊂ IRn . The function f is called a d.c. function on A if it can be expressed as a difference of two convex functions on A, i.e., there exist convex functions f1 and f2 on A such that f (x) = f1 (x) − f2 (x) ∀x ∈ A. The pair of functions (f1 , f2 ) is said to be a d.c. representation of f on A. Moreover, the functions f1 and f2 are called the first component and the second component respectively of the current d.c. representation of f on A. On the other hand, a function f is said to be d.c. at a point x ∈ A if there exists a convex neighborhood Ux of x such that f is d.c. on Ux ∩ A. If f is d.c. at every point of A, it is said to be locally d.c. on A. Every locally d.c. function on A ⊂ IRn , A an open or closed convex set, is d.c. on A (see Hartman [25]). Moreover, it can be proved that every function f ∈ C 2 (A), A open or closed convex set, is a d.c. function on A (see Ellaia [14]). The set of the d.c. functions on A, denoted by DC(A), is the vector space generated by the cone of convex functions on A. Given f ∈ DC(A), it is evident that there are infinitely d.c. representations of f . Denote by Df (A) the set Df (A) := {(f1 , f2 ) : f (x) = f1 (x) − f2 (x) ∀x ∈ A, f1 , f2 convex on A}. When a d.c. representation (f1 , f2 ) of f is available, then we can always obtain a new d.c. representation of f in the form (f1 + g, f2 + g), where both components are strictly convex by adding a strictly convex function g(x) (a simple choice is g(x) = tkxk2 with t > 0). DC(A) has some interesting properties with respect to operations frequently encountered in optimization (see Hiriart-Urruty [27] or Horst et al [32])..

(25) xix. PRELIMINARIES. • Every linear combination of a finite number of d.c. functions is a d.c. function, which is a consequence of well known properties of convex and concave functions. • Let (pi , qi ), i = 1, . . . , m be d.c. representations of the functions fi , i = 1, . . . , m. Then, max{f1 (x), . . . , fm (x)} is a d.c. function because we can write  . max{f1 (x), . . . , fm (x)} = max pi (x) + . m X. j=1, j6=i.  m  X. qj (x), i = 1, . . . , m −. • min{f1 (x), . . . , fm (x)} is a d.c. function because we know. . qj (x).. j=1. min{f1 (x), . . . , fm (x)} = −max{−f1 (x), . . . , −fm (x)}. • Also, we can see that |f (x)| := max{f (x), −f (x)}, f + (x) := max{0, f (x)} and f − (x) := min{0, f (x)} are d.c. functions. • The product of a pair of nonnegative-valued convex functions q1 and q2 is a d.c. function because we can write q1 (x)q2 (x) =. 1 1 (q1 (x) + q2 (x))2 − (q12 (x) + q22 (x)). 2 2. • Let f1 and f2 be d.c. functions on A, A open or closed convex set in IRn . Then, f1 (x)f2 (x))) and, if for all x ∈ A, f2 (x) 6= 0, the quotient f1 (x)/f2 (x))) are d.c. functions on A (see Hartman [25]). Some authors provide interesting theoretical d.c. representation results but no practical means to get them. D.c. functions were considered by Alexandrov [1] and Landis [44]. Some time later Hartman [25] states that every locally d.c. function on A ⊂ IRn , A open or closed convex set, is d.c. on A. Bougeard [6] proves that if f ∈ C 2 (A) then there exists a d.c. representation (f1 , f2 ) of f in which f1 ∈ C 2 (A) and f2 ∈ C ∞ (A). Penot and Bougeard [50] establish a similar result with more global assumptions. Indeed, let A be an open convex set of a finite dimensional normed vector space, then any lower−C 2 function f on A, in particular any f ∈ C 2 (A)), can be written as f = f1 − f2 with f1 and f2 convex and f2 ∈ C ∞ (A). Moreover, every lower−C 2 function can be characterized by its (locally) decomposability as a sum of a convex continuous and a concave quadratic function (see [77])..

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(27) Introduction Motivation A global optimization programming problem has the general form minimize f (x) subject to: x ∈ S, where S is a set contained in IRn and the minimizer is understood in the global sense, i.e., we are interested in points x∗ ∈ S satisfying f (x∗ ) ≤ f (x) ∀x ∈ S. The set of global minimizers is denoted by argminf (S) and, at each x∗ ∈ argminf (S), the corresponding value f (x∗ ) is said to be the global minimum of the function f at the point x∗ over the set S. On the other hand, a point xo ∈ S is said to be a local minimizer of the function f over the set S, if there exists a neighborhood V of xo satisfying f (xo ) ≤ f (x) ∀x ∈ S ∩ V. The corresponding value f (xo ) is said to be the local minimum of the function f at the point xo over the set S. If the set S can be described as S := {x : gi (x) ≤ 0, i = 1, . . . , m} and if all functions involved in the program are in C 1 (A), A an open set containing S, the following Karush-Kuhn-Tucker (KKT ) conditions hold at xo ∈ S. There exist λi ≥ 0, i = 1, . . . , m, such that 1. λi gi (xo ) = 0, i = 1, . . . , m, 2. ∇f (xo ) +. Pm. i=1 λi ∇gi (x. o). = 0, 1.

(28) 2. INTRODUCTION. provided that the constraints at the local minimizer xo are regular. When both the objective function f and the feasible set S are convex the program is said to be convex and, in this case, it is well known that every local minimizer is a global one. Suppose that by using a standard local technique of nonlinear programming we have obtained a Karush-Kuhn-Tucker point of the above program. We must then stop the procedure because no local method can tell us whether the obtained point is a global optimizer or not and, in the latter case, how to proceed to obtain a better feasible point. It is this possibility of becoming trapped at a stationary point that causes the failure of local methods and motivates the need to develop global ones. In practice, direct problem formulations are not convenient, and it is thus necessary to transform them into alternative ones that are more suitable for algorithmic purposes. In any approach to global optimization methods it is essential to understand the mathematical structure of the problem under consideration. A careful analysis of this structure can provide insight into the most relevant properties of the problem and suggest efficient methods for solving it. In recent years, some papers have described deterministic global optimization procedures to solve problems whose objective function can directly be expressed as difference of convex functions (d.c. function), and the feasible domain is a convex set. They are a special class of global optimization programs named d.c. programs which are described in Subsection 1.4.3. The Multisource Weber Problem, the Facility Location Problem with limited distances, the Stochastic Transportation-Location Problem and the Stochastic Transportation Problem belong to this special class of d.c. programs. In Pey-Chun Chen et al [51], both the Multisource Weber Problem and the Facility Location Problem with limited distances are reformulated as a concave minimization problem, which is the simplest class of global optimization problem. In K. Holmberg et al [30], the Stochastic Transportation-Location Problem and the Stochastic Transportation Problem are reduced to d.c. optimization problems whose objective functions are separable d.c. functions and the feasible domains are defined by the transportation constraints. In this case, the procedure suggested in [30] takes advantage of these two special structures so an efficient rectangular subdivision branching method can be used to solve them. This Thesis has been motivated by the interest in applying deterministic global optimization procedures to problems in the real world with no special structure. We have focused on the Short-Term Hydrothermal Coordination of Electricity Generation Problem (also named the Generation Problem in this Thesis) where the objective function and the nonlinear constraints are polynomials of degree up to four (see [26]). Its solution has important economic and technical implications. In the.

(29) 3. INTRODUCTION. Generation Problem neither a representation of the involved functions as difference of convex functions is at hand nor we can take advantage of any special structure of the problem. Hence, a very general problem such as minimize f (x) subject to: x ∈ S ⊂ IRn , does not seem to have any mathematical structure conducive to computational implementations. However, when f (x) is a continuous function and S is a nonempty closed set the problem can be transformed into an equivalent problem expressed by minimize l(z) subject to: z ∈ D \ intC, which is said to be a canonical d.c. program, where l(z) is a linear function and D and C are closed convex sets (see Section 1.3 for details). Thus, we can see that every continuous global optimization problem has a mathematical complementary convex structure (D \intC) also called the d.c. structure. The mathematical complementary convex structure is not always apparent and, even when it is explicit, a lot of work still remains to be done to bring it into a form amenable to efficient computational implementations. The attractive feature of the mathematical complementary convex structure is that it involves convexity. Thus, we can use analytical tools from convex analysis like subdifferential and supporting hyperplane. On the other hand, since convexity is involved in the reverse convex property, these tools must be used in some specific way and combined with combinatorial tools like cutting planes, branch and bound and outer approximation.. Objectives A program expressed by minimize f (x) subject to: hi (x) ≤ 0, i = 1, . . . , m, x ∈ S, is said to be a d.c. program when S is a closed convex set into IRn and the functions f (x) and hi (x), i = 1, . . . , m are d.c. functions, i.e., they are expressed explicitly as a difference of two convex functions on S. At the expense of introducing additional variables, any d.c. program can be transformed to a program with a complementary convex structure. While it is not too difficult to prove theoretically that a given.

(30) 4. INTRODUCTION. function is a d.c. function, it is often very problematical to obtain an effective d.c. representation of a d.c. function as a difference of convex functions. For this reason our first objective is O1.- how to write effectively a function as a difference of convex functions. After that, our second objective is O2.- to convert the Generation Problem into an equivalent reverse convex programming problem (see Section 1.4 for details) and develop a deterministic global optimization procedure to solve it. Having solved the problem of finding a d.c. representation of a polynomial we then come up against another even more complicated problem, that is, if the computational efficiency depends on the d.c. representation of the functions. Our third objective is to answer the questions: O3.- is there any d.c. representation that improves the computational efficiency? If the answer to this question is affirmative, then what is the best d.c. representation of a d.c. function (optimal d.c. representation) from a computational point of view and how can it be obtained? Finally, O4.- we want to compare, for the nonconvex Generation Problem, the solutions obtained by applying the deterministic global optimization algorithm and the solutions obtained with a local optimization package. This comparison will shed light on two topics: – how far the solutions obtained by the local optimization package are from the global optimizer, and – up to which problem size the global procedure developed can be applied in practice..

(31) INTRODUCTION. 5. Contributions In this Thesis, we have O1a.- described a new method for obtaining a d.c. representation of polynomials based on the fact that the set of mth powers of homogeneous polynomials of degree 1 is a generating set for the vector space of homogeneous polynomials of degree m, O1b.- developed a procedure, using M AP LE Symbolic Calculator, which allows us to search for bases and to obtain a d.c. representation of the homogeneous components of a polynomial. Alternative bases in order to obtain a different d.c. representation of a polynomial can be used,. O2a.- described and written out a procedure in F ORT RAN to convert the Generation Problem into an equivalent reverse convex programming problem expressed by: minimize f (x) − t subject to: g(x) − t ≤ 0, h(x) − t ≥ 0, Ax ≤ b, where A is a real m × n matrix, b ∈ IRm and f (x), g(x) and h(x) are convex functions on IRn , O2b.- described and written out an adapted algorithm in F ORT RAN and in C by modification of the combined outer approximation and cone splitting conical algorithm for canonical d.c. programming from [74]. Since the above-mentioned programming problem is unbounded we use prismatical subdivisions instead of conical ones so that it is not necessary to find a subdivision vertex as in the case of conical subdivisions in [74]. Moreover, the adapted algorithm uses prismatic branch and select technique with polyhedral outer approximation subdivisions, in such a way that only linear programming problems have to be solved. To solve them, we have used the M IN OS package in the case of the algorithm in F ORT RAN and the CP LEX callable library in the case of the algorithm in C. O2c.- established theoretically the convergence to a global optimizer of the adapted algorithm,.

(32) 6. INTRODUCTION. O3a.- applied the concept of least deviation decomposition from [46] to obtain an optimal d.c. representation of a polynomial function in the normed space of the polynomials with the Euclidean norm, in order to improve the computational efficiency of our algorithm. O3b.- described and written out an algorithm in F ORT RAN by using an interior point method to solve semi-infinite quadratic programming problems with linear constraints to obtain these optimal d.c. representation (see [11], [37] and [78] for more details), O4a.- used M IN OS to check all gradients of the d.c. program instances, O4b.- used M IN OS as a local optimization package to compare its solutions with the solutions of the adapted algorithm. It should be pointed out that the adapted algorithm is more general than it seems to be because it can be used to solve d.c. programming problems with convex constraints expressed by minimize f (x) − g(x) subject to: hi (x) ≤ 0, i = 1, . . . , m, which, by introducing an additional variable t, can be transformed into an equivalent convex minimization problem subject to an additional reverse convex constraint in the form minimize f (x) − t subject to: g(x) − t ≥ 0, h(x) ≤ 0, with h(x) = max{hi (x) :, i = 1, . . . , m}. This is a different way of solving a d.c. program with convex constraints than the algorithm proposed in [34] and [33] which transforms it into an equivalent concave minimization problem and uses prismatic branch and bound technique with polyhedral outer approximation subdivisions.. Contents Chapter 1 begins with a brief historical note about optimization methods, in which deterministic global optimization has a history of over thirty years. Then, we introduce the common general mathematical complementary convex structure underlying in global optimization problems. In this Thesis, this general mathematical complementary convex structure provides the foundation for reducing a nonconvex global.

(33) INTRODUCTION. 7. optimization problem (for which a d.c. representation of both the objective function and constraints can be obtained) to a canonical form. Moreover, any global optimization method must address, directly or indirectly, the question of how to transcend a given feasible solution, which may be a local minimizer or an stationary point. It is this possibility to be trapped at an stationary point which causes the failure of local methods. Hence, various methods, which have been proposed to solve global optimization problems, are briefly mentioned. In Chapter 2 we describe the Generation Problem, whose functions are d.c. functions because they are polynomials. Thus, by using the properties of the d.c. functions (see [27] and [32]) and the flow balance equations at all nodes of the replicated hydronetwork (which are the linear constraints of the Generation Problem), we describe the Generation Problem as an equivalent canonical d.c. programming problem of reduced size. It should be stressed that several transformations can be used to obtain an equivalent reverse convex program. From the structure of the functions in the Generation Problem, we rewrite it as a more suitable equivalent reverse convex program in order to obtain an advantageous adaptation of the combined outer approximation and cone splitting conical algorithm for d.c. programming as described in [74]. Chapter 3 introduces the concepts and properties, which allow us to obtain an explicit representation of a polynomial as a difference of convex polynomials (Corollary 3.3.4), based on the fact that the set of mth powers of homogeneous polynomials of degree 1 is a generating set for the vector space of homogeneous polynomials of degree m (Proposition 3.3.2). Also, we compare our procedure to obtain a d.c. representation of a polynomial with the procedure described by Konno, Thach and Tuy [42], emphasizing its advantages and applying it to the polynomials of the Generation Problem. Moreover, we present a procedure, using M AP LE Symbolic Calculator, which allows us to search for bases and to obtain a d.c. representation of the homogeneous components of a given polynomial. Chapter 4 is devoted to describing the adapted global optimization algorithm and its basic operations. Moreover, we prove the convergence of the adapted algorithm by using a prismatical subdivision process together with an outer approximation procedure. The adapted global optimization algorithm is an advantageous adaptation of the combined outer approximation and cone splitting conical algorithm for d.c. programming in [74]. Since our equivalent reverse convex program is unbounded we use prismatical subdivisions instead of conical ones (as used in [74]). Hence, it is not necessary to find any subdivision vertex as in the case of conical subdivisions..

(34) 8. INTRODUCTION. In Chapter 5, we announce the minimal norm problem by using the concept of least deviation decomposition (LDD) described in Luc D.T. et al [46] in order to obtain the optimal d.c. representation of a polynomial function, which allow us more efficient implementations by reducing the number of iterations of the adapted global optimization algorithm. The minimal norm problem can be transformed into an equivalent semidefinite program or into an equivalent semi-infinite quadratic programming problem with linear constraints. We discuss the suitability of use a quadratic semi-infinite algorithm in place of a semidefinite one. Chapter 6 is devoted to describing a quadratic semi-infinite algorithm, which is an adaptation of the linear semi-infinite algorithm developed by J.Kaliski et al in [37], and its basic operations. We propose a build-up and build-down strategy, introduced by Den Hertog in [11] for standard linear programs that use a logarithmic barrier method. It should be pointed out that Chapters 5 and 6 are closely connected. They are presented separately for the sake of clarity. Finally, in Chapter 7 computational results are given and conclusions are discussed..

(35) Chapter 1 Overview and scope of global optimization. 1.1. Historical notes. Modern techniques of optimization constitute a scientific discipline whose origins can be traced back to the birth of the digital computer. Optimization is concerned with the analysis of solutions and the development of procedures for solving problems of the form minimize f (x) (1.1) subject to: gi (x) ≤ 0, i = 1, 2, . . . , m n x ∈ X ⊂ IR , where f (x) and gi (x), i = 1, 2, . . . , m, are real-valued functions defined on a domain X ⊂ IRn . Hence, f (x) is the objective function which measures the quality of the solution, S := {x ∈ X : gi (x) ≤ 0, i = 1, 2, . . . , m} is the set of feasible points (or feasible domain), and x ∈ S are said to be the decision variables. If m = 0 and S defines an hyperrectangle of IRn , the problem is said to be unconstrained; if n = 1 the problem is univariate otherwise the problem is said to be multivariate. Linear programming deals with optimization problems where f (x) is a linear function and the feasible domain S is a set defined by linear inequalities. During World War II, George B. Dantzig developed the simplex method for solving linear programming models of logistics and operational military problems. At the same time, and independently, Leonid V. Kantorovich developed the method of resolving multipliers for linear programs and applied them to problems such as equipment work distribution. The simplex method consists of the selection of an optimizer 9.

(36) 10. CHAPTER 1. OVERVIEW AND SCOPE OF GLOBAL OPTIMIZATION. within the finite set of vertices of the convex polytope defined by the linear constraints. A programming problem is said to be combinatorial when an optimizer must be sought within a finite set of candidate points. Integer programming imposes additional integrality constraints on some subsets of the decision variables. Almost all combinatorial optimization problems can be modelled as integer programs. Methods such as cutting-plane, branch and bound, branch and cut, column generation, decomposition techniques and polyhedral combinatorics have served as powerful tools for solving integer programming problems. However, because of the complexity (NP-hard, etc) of these problems, many instances cannot be solved exactly in a polynomial time. Methods which find approximate optimizers (suboptimal solutions) to these instances have been developed in recent times. Heuristics, such as genetic algorithms by Holland in 1975 [29], simulated annealing by Kirkpatrick et al. in 1983 [41], GRASP by Feo and Resende in 1995 [15], Tabu search by Glover and Laguna in 1997 [22], find good quality approximate optimizers in reasonable computational times. Network optimization is another important field of combinatorial optimization. Many early network algorithms, such as minimum spanning tree by Kruskal in 1956 [43], Prim in 1957 [54] and shortest-path programs by Dijkstra in 1959 [12] are still used today. In 1962 the book Flows in networks by Ford and Fulkerson [19] appeared. Recent data structure developments have contributed to the use of network algorithms for solving large real-world optimization problems. In nonlinear optimization the constraints, which define the feasible domain S, and/or the objective function f (x) are nonlinear functions. In the early 1960’s, J.B. Rosen published the gradient-projection method for nonlinear programming (see [58] and [59]) which instigated further research into algorithms for nonlinear optimization. The interplay between continuous nonlinear optimization and combinatorial optimization motivated the development of new algorithmic techniques for largescale problems. For instance, the field of interior point methods uses nonlinear programming for solving linear programs. In 1979 Leonid Khachiyan published the ellipsoid method [39], the first polynomial time algorithm for solving linear programs which come from nonlinear programming was published in 1970 by Shor ([67] and [66]), and it is drastically different from most previous approaches to linear programming. In 1984 Narendra Karmarkar published a polynomial time interior point algorithm for linear programming [38], which is the origin of modern interior point methods. Variants of interior point methods were shown to perform well in practice and they extended the limits of the dimension of problems that could be solved. When a problem is convex, it is well known that every local minimizer is global. In many practical problems in which the convexity of the objective function or the.

(37) 11. 1.1. HISTORICAL NOTES. Figure 1.1: f (x, y) :=. 3 2 100 (x. + y 2 ) − cos(x)cos(y). constraints cannot be verified, it is reasonable to assume that the problems have multiple local optimizers (multiextremal problems, see Figure 1.1). Global optimization deals with the computation and characterization of global optimums of multiextremal problems and it aims at solving the very general program (1.1). Any global optimization method must address the question of how to transcend a given feasible point, if there is one, or else how to produce evidence that the given point is already globally optimal. Global optimization techniques are substantially different from local ones and, among others, employ combinatorial tools such as cutting-plane, branch and bound, branch and cut and so on. Most people considered until the mid 1980’s that heuristic and stochastic local searches were more practical and reliable approaches for solving these inherently difficult classes of problems. In recent years, despite the inherent difficulty of global optimization, remarkable developments in this field have been made. The emergency of powerful workstations enabled one to solve a number of small to medium size global optimization problems by general purpose deterministic algorithms in a practical amount of time. They have been applied to some important classes of problems such as concave minimization, reverse convex programs, d.c. programs and Lipschitz optimization. Unfortunately we usually observe a rapid increase in computation time as the size of the instance increases if it has no special structure. The first textbook on this subject was published in 1990 by Horst and Tuy [32]. In this textbook the authors discussed the overall theoretical framework and general purpose deterministic algorithms for locating a global optimum of a multiextremal problem. The last years also witnessed the emergence of the Journal of Global Optimization as well as the increase in research activities with several specialized conferences on global optimization and applications..

(38) 12. CHAPTER 1. OVERVIEW AND SCOPE OF GLOBAL OPTIMIZATION. 1.2. Classification of global optimization methods. As we have seen before any global optimization method must address the question of how to transcend a given feasible point or else to produce evidence that the given point is already globally optimal. Numerical methods that have been proposed can be classified into three categories by distinguishing the available guarantees: heuristic methods, approximation methods, and systematic methods, which include deterministic methods.. 1.2.1. Heuristic methods. Heuristic methods are used for solving global optimization problems whose structure is difficult to analyze. They contain all methods in which no theoretical justification can be established to find a global optimizer. In these methods, we have some grounds for believing that the feasible point obtained is sufficiently near the optimum or that there is a high probability of it being so. Among the most popular are: • Multi-start methods which are intended to locate as many local minimizers as possible by applying standard local optimization algorithms initiated at multiple starting points which are generated stochastically. • Simulated annealing methods which are intended to transcend a local solution by using some probabilistic procedure which may accept a temporary increase in the objective function. • Genetic algorithms where the optimizer is searched by a procedure imitating the natural genetic selection of the fittest individuals. Also, heuristic algorithms of tabu search type have been used to solve global optimization problems difficult to tackle otherwise. In contrast to deterministic methods, heuristic methods offer almost no guarantee of global optimality for the obtained solution and hardly provide any information about how far this solution could be from the optimum.. 1.2.2. Approximation methods. Approximation methods transform the original problem by means of suitable approximations into a simpler global optimization problem that is more tractable. Solving.

(39) 1.2. CLASSIFICATION OF GLOBAL OPTIMIZATION METHODS. 13. the approximate problem yields an approximate solution for the original problem, and local optimization from this approximate solution often gives the global optimizer of the original problem provided that the approximation was good enough or a good local minimizer otherwise. Of great practical importance is the approximation by the mixed integer linear programs (MILP).. 1.2.3. Systematic methods. When only black box function evaluation routines are available, we can use some kind of systematic methods (black box methods) which will usually find a global optimizer with certainty in a finite time, but we can never know when this is the case. Thus, for systematic black box methods, stopping must be based on heuristic recipes. On the other hand, deterministic methods contain all methods that theoretically guarantee a global optimizer via a predictable amount of work. Predictable means relative to known problem characteristics such as a d.c. structure, Lipschitz constants or other global information needed for the convergence proof. These methods include, among others, cutting plane, successive approximation and partitioning methods together with concepts such as concavity cuts, outer approximation schemes and branch and bound techniques. However, a weakness of these methods is that for problems without any particular structure, they can only solve, in practice, problem instances of limited size. The amount of work is usually exponential with respect to the problem characteristics. Sometimes, heuristic and probabilistic choices also play a role in systematic methods, mainly to provide cheaply a good local optimizer which benefits the systematic search. We list some systematic global optimization codes with short comments on scope and method. The codes listed use black box function evaluation routines, and have heuristic stopping rules. DIRECT (Divide Rectangles). It is a branching code in F ORT RAN which use function values only, by Gablonsky and Kelley (2001). DIRECT is based on branching and a Pareto principle for box selection (see [20]). http://www4.ncsu.edu/∼jmgablon/. MCS (Multilevel Coordinate Search). A M AT LAB program for bound constrained global optimization by Huyer and Neumaier (1999). M CS uses function values only and is based on branching and sequential quadratic program-.

(40) 14. CHAPTER 1. OVERVIEW AND SCOPE OF GLOBAL OPTIMIZATION. ming (see [35]). http://solon.cma.univie.ac.at/∼neum/software/mcs/. LGO (Lipschitz Global Optimization). An integrated development environment for global optimization with Lipschitz continuous objective function and constraints by Janos Pintér (1996). LGO is based on branching and estimation of Lipschitz constants, constraints other than simple bounds are handled by L1 penalty terms and interior convex constraints by projection penalties (see [52]). http://is.dal.ca/∼jdpinter/lgoide.html/.. The following codes use global information and stop in a finite time with the guarantee that the global minimizer is found. In difficult cases storage or time limits may be exceeded, however, leading to appropriate error messages.. BARON (Branch and Reduce Optimization Navigator). A general purpose solver for optimization problems with nonlinear constraints and/or integer variables by Ryoo and Sahinidis (1996). BARON is based on branching and box reduction using convex relaxation and Lagrange multiplier techniques (see [62]). http://archimedes.scs.uiuc.edu/baron/baron.html. αBB (alpha Branch and Bound). A branch and bound code for nonlinear programs by Androulakis, Maranas and Floudas (1995). It is based on branching and bound by convex underestimation using interval analysis to write nonlinearities in difference of convex functions form (see [3]). http://titan.princeton.edu/soft.html#abb. The site contains a description only, no code.. Recently, the Cutting Angle Method of Andramonov, Rubinov et al. (see [2] and [61]) has been developed for solving global optimization problems. A new version of this method can be found in [5]. In this method the objective function f (x) is a Lipschitz function and the feasible domain S is the unit simplex. Many problems of unconstrained and constrained optimization can be reduced to this form by a transformation of variables and penalization. In this Thesis we are interested in deterministic global optimization methods which rely on a d.c. structure of a d.c. program. These methods have been proved to be particularly successful for analyzing and solving a variety of highly structured problems (see [30] and [51])..

(41) 15. 1.3. GENERAL COMPLEMENTARY CONVEX MATHEMATICAL STRUCTURE. 1.3. General complementary structure. convex. mathematical. A general problem such as (1.1) does not seem to have any mathematical structure receptive to computational implementations. However, by taking into account the fact that every point in IRn has a base of convex neighborhoods, when f (x) is a continuous function and S is a nonempty closed set the problem (1.1) can be transformed into an equivalent problem with a linear objective function and a complementary convex mathematical structure (or d.c. structure) expressed by minimize l(z) subject to: z ∈ D \ int C,. (1.2). where l(z) is a linear function and D and C are closed convex sets. The program (1.2) is said to be a canonical d.c. program. For the convenience of making the exposition self-contained we quote some results with proofs from Tuy [74]. Proposition 1.3.1 Let S ⊂ IRn be a closed convex set. Then for every y ∈ IRn \ S, let B(y, δy ) ⊂ IRn \ S be a ball with center y and radius δy > 0. The following equalities are true: S=. T. y 6∈ S. IRn \ B(y, δy ) = {x ∈ IRn : g(x) − kxk2 ≤ 0},. where g(x) := sup {2xy − kyk2 + δy2 , y 6∈ S} is a convex lower semi-continuous function ( l.s.c.) (see [56]). Proof: For all y 6∈ S there exists a ball B(y, δy ) of radius δy > 0 satisfying B(y, δy ) ⊂ IRn \ S or equivalently S ⊂ IRn \ B(y, δy ). Then S ⊂ ∩y6∈S IRn \ B(y, δy ). Moreover, if x ∈ ∩y6∈S IRn \ B(y, δy ) and x 6∈ S, then x 6∈ B(x, δx ) which is a contradiction. So ∩y6∈S IRn \ B(y, δy ) ⊂ S. On the other hand, x ∈ S ⇔ ky − xk ≥ δy , for all y 6∈ S. Then, we have ky − xk2 = kyk2 + kxk2 − 2xy ≥ δy2 , for all y 6∈ S followed by and kxk2 ≥ 2xy − kyk2 + δy2 , for all y 6∈ S 2 2 2 kxk ≥ g(x) = sup {2xy − kyk + δy , y 6∈ S}, where g(x) := sup {2xy − kyk2 + δy2 , y 6∈ S} is the pointwise supremum of a family of affine functions. Thus, x ∈ S ⇔ g(x) − kxk2 ≤ 0..

(42) 16. CHAPTER 1. OVERVIEW AND SCOPE OF GLOBAL OPTIMIZATION. Proposition 1.3.2 Let S ⊂ IRn be a closed set. There exist closed convex sets D and C into IRn+1 such that S is the projection of D \ int C on IRn , i.e., S = projIRn (D \ int C). Proof: Using the results of the Proposition 1.3.1 we have that g(x) − kxk2 ≤ 0 ⇔ there exists t ∈ IR with g(x) − t ≤ 0 and kxk2 − t ≥ 0. Thus, D = {(x, t) ∈ IRn+1 : g(x) − t ≤ 0} and C = {(x, t) ∈ IRn+1 : kxk2 − t ≤ 0} are closed convex sets and verify S = projIRn (D \ int C).. Proposition 1.3.3 Problem (1.1), with continuous objective function and closed feasible set, can be transformed into an equivalent canonical d.c. program expressed by minimize l(z) subject to: z ∈ D \ int C, where l(z) is a linear function and D and C are closed convex sets. Proof: The program (1.1) can be expressed by the equivalent programming problem minimize t subject to: f (x) − t ≤ 0, x ∈ S, (1.3) where {(x, t) ∈ IRn+1 : f (x) − t ≤ 0, x ∈ S} is a closed set. From Proposition 1.3.2 {(x, t) ∈ IRn+1 : f (x) − t ≤ 0, x ∈ S} = projIRn+1 (D \ int C) with D and C closed convex sets into IRn+2 . We obtain the expression (1.2) by setting z = (x, t). Hence, we can see that every continuous global optimization problem can be expressed by an equivalent canonical d.c. program. A canonical d.c. program is said to be regular when the feasible set satisfies the property D \ int C = cl (D \ C). Also, a canonical d.c. program is said to be essentially nonconvex when it satisfies the inequality min{l(x) : x ∈ D} < min{l(x) : x ∈ D \ int C}.. (1.4). Denote D(γ) := {x ∈ D : l(x) ≤ γ} with γ ∈ IR. Assume the problem (1.2) to be regular and essentially nonconvex. Let x̄ ∈ D \ int C be a feasible point and set γ̄ = l(x̄). Then, we can announce the propositions.

(43) 1.3. GENERAL COMPLEMENTARY CONVEX MATHEMATICAL STRUCTURE. 17. Proposition 1.3.4 If x̄ ∈ D \int C is a global optimizer of (1.2) then x̄ ∈ ∂C ∩∂D. Proof: Suppose that x̄ 6∈ ∂C, then x̄ 6∈ C and x̄ ∈ D \ C = D ∩ IRn \ C that is an open set by the topology induced in D from the topology in IRn . Hence, there exists a ball B(x̄, δ) such that B(x̄, δ) ∩ D ⊂ D \ C and if 0 < t ≤ δ/kx̄ − wk < 1. From the convexity of the function l(x) every point z = (1 − t)x̄ + tw satisfies l(z) ≤ (1 − t)l(x̄) + tl(w) = l(x̄) + t(l(w) − l(x̄)) < l(x̄), which is a contradiction. On the other hand, suppose that x̄ 6∈ ∂D. From x̄ ∈ ∂C every ball B(x̄, δ) satisfies B(x̄, δ) ∩ D(γ) ∩ int D 6= ∅ and B(x̄, δ) ∩ D(γ) ∩ ext D 6= ∅. Consider z0 ∈ B(x̄, δ) ∩ D(γ) ∩ ext D. Then every point z = (1 − t)z0 + tw with 0 ≤ t ≤ 1 satisfies l(z) ≤ (1 − t)l(z0 ) + tl(w) ≤ (1 − t)l(x̄) + tl(w) = l(x̄) + t(l(w) − l(x̄)) < l(x̄), by using the convexity of the function l(x). Hence, as z0 ∈ D \ C and w ∈ D ∩ C there exists z̃ ∈ ∂C ∩ [w, z0 ] satisfying l(z̃) < l(x̄) which is a contradiction.. Proposition 1.3.5 The following assertions are true: i) if D(γ̄) \ C 6= ∅, then there exists a feasible point strictly better than x̄ lying on ∂C ∩ D, ii) x̄ is a global optimizer of (1.2) ⇔ D(γ̄) \ C = ∅ (or D(γ̄) ⊂ C). Proof: ii) ⇒(necessary) Let x0 ∈ D(γ̄) \ C. Then, l(x0 ) = l(x̄) = γ̄ so that x0 is an optimal global solution of (1.2). Thus, x0 ∈ ∂C (i.e. x0 ∈ C) is a contradiction. ⇐(sufficient) Let x∗ ∈ D \ int C be a better feasible solution than x̄. Then, l(x∗ ) < l(x̄) and we can find an open ball B := B(x∗ , δ), of center x∗ and radius δ, such that for all x ∈ B we have 1) l(x) < l(x̄). On the other hand, from x∗ ∈ D \ int C = cl(D \ C) it follows that B ∩ (D \ C) 6= ∅ so every point x ∈ B ∩ (D \ C) verifies 2) x ∈ D and x 6∈ C. Thus, from 1) and 2) we have x ∈ (D(γ̄) \ C) = ∅ which is a contradiction..

(44) 18. 1.4. CHAPTER 1. OVERVIEW AND SCOPE OF GLOBAL OPTIMIZATION. Classes of nonconvex problems with a d.c. structure. In many classes of global optimization problems, convexity is present in a reverse sense. In this circumstance we have minimization of concave functions subject to convex constraints (concave minimization), minimization of convex functions on feasible domains which are the intersection of convex sets and complements of convex sets (reverse convex programming) and global optimization of functions which can be expressed as a difference of two convex functions (d.c. programming).. 1.4.1. Concave minimization. A program expressed by minimize f (x) subject to: x ∈ S,. (1.5). where f (x) is a concave function on IRn and S ⊂ IRn is a nonempty closed convex set, is said to be a concave minimization program. This is the simplest class of global optimization problems with a d.c. structure. By defining the closed convex sets D := {(x, t) ∈ IRn+1 : x ∈ S} and C := {(x, t) ∈ IRn+1 : f (x) ≤ t}, it is equivalent to the canonical d.c. program expressed by minimize t subject to : (x, t) ∈ D \ int C.. 1.4.2. (1.6). Reverse convex programming. A program expressed by minimize f (x) subject to: g(x) ≤ 0, h(x) ≥ 0. (1.7). where f (x), g(x) and h(x) are convex functions on IRn is said to be a reverse convex program. This differs from a convex program only by the presence of the constraint h(x) ≥ 0, which is said to be a reverse convex constraint. The reverse convex program (1.7) is equivalent to the canonical d.c. program expressed by minimize t subject to: (x, t) ∈ D \ int C,. (1.8). where D and C are closed convex sets into IRn+1 defined by D := {(x, t) ∈ IRn+1 : f (x) − t ≤ 0, g(x) ≤ 0} and C := {(x, t) ∈ IRn+1 : h(x) ≤ 0}..

(45) 1.4. CLASSES OF NONCONVEX PROBLEMS WITH A D.C. STRUCTURE. 1.4.3. 19. D.c. programming. A program expressed by minimize f (x) subject to: x ∈ S, hi (x) ≤ 0, i = 1, . . . , m,. (1.9). is said to be a d.c. program when S is a closed convex set into IRn and the functions f (x), hi (x) ≤ 0, i = 1, . . . , m are d.c. functions and they can be explicitly expressed as a difference of two convex functions on IRn . By introducing the additional variable t ∈ IR the program (1.9) can be transformed to the equivalent d.c. program minimize t subject to: x ∈ S, f (x) − t ≤ 0, hi (x) ≤ 0, i = 1, . . . , m,. (1.10). with linear objective function. Furthermore, the d.c. inequalities f (x) − t ≤ 0, hi (x) ≤ 0, i = 1, . . . , m, can be replaced by a single d.c. inequality r(x, t) := max {f (x) − t ≤ 0, hi (x) ≤ 0, i = 1, . . . , m} ≤ 0. Suppose that for each d.c. function f (x), hi (x) ≤ 0, i = 1, . . . , m a d.c. representation is known. Then, by using the properties of the d.c. functions (see Preliminaries) we can write r(x, t) = p(x, t) − q(x, t), where p(x, t) and q(x, t) are convex functions. Hence, by introducing the new additional variable z ∈ IR, the d.c. inequality r(x, t) = p(x, t) − q(x, t) ≤ 0 is equivalent to the system p(x, t) − z ≤ 0, q(x, t) − z ≥ 0, where the first inequality is convex and the second is said to be reverse convex. Finally, setting D := {(x, t, z) ∈ IRn+2 : p(x, t) − z ≤ 0, x ∈ S} and C := {(x, t, z) ∈ IRn+2 : q(x, t) − z ≤ 0}, we can see that the d.c. program (1.9) is equivalent to the canonical d.c. program minimize t subject to: (x, t, z) ∈ D \ int C.

(46) 20. 1.4.4. CHAPTER 1. OVERVIEW AND SCOPE OF GLOBAL OPTIMIZATION. Continuous optimization. It has been proved in Proposition 1.3.3 that the problem of minimizing a continuous function over a closed set can be reduced to a canonical d.c. program. This is a theoretical result and, in any case, continuous optimization problems are the most difficult to solve from the viewpoint of the complementary mathematical convex structure. As above-mentioned, in this Thesis we are interested in deterministic global optimization methods which rely on a d.c. structure of a d.c. program. The main question is how to find a good d.c. structure of a given d.c. program. We will solve this problem when the functions of the d.c. program are polynomial functions..

(47) Chapter 2 The Generation Problem and its complementary convex structure. 2.1. The short-term hydrothermal coordination of electricity generation problem. We want to apply deterministic global optimization procedures to the problem of the short-term hydrothermal coordination of the electricity generation (Heredia and Nabona [26]). Its importance stems from the economic and technical implications that the solution to this problem has for electric utilities with a mixed, hydro and thermal, generation system. Given a short-term time period subdivided into time intervals the aim is to find values of hydro and thermal generation for each time interval in the period so that the demand of electricity is met for each time interval, a number of constraints are satisfied, and the generation cost of thermal units is minimized. The model contains the replicated hydronetwork through which the temporary evolution of the reservoir system is represented (Figure 2.1 shows us the replicated hydronetwork with only two reservoirs and where the time period has been subdivided into four time intervals). We use Ne to indicate the number of reservoirs, Nt to indicate the number of time intervals, j to indicate the j th reservoir, j = 1 . . . Ne and i to indicate the ith time interval, i = 1 . . . Nt . It should be observed that, • the variables are the water discharges dij from reservoir j over the ith interval and the volume stored v i in reservoir j at the end of the ith time interval, j. 21.

(48) 22. CHAPTER 2. THE GENERATION PROBLEM AND ITS COMPLEMENTARY CONVEX STRUCTURE. 1. 2. w1. 1. v1. R1. 3. w1. 2. v1. R1. 4. w1. w1. 3. v1. R1. 4. v1. R1. 0. v1 1. d1. 1. w2. 2. w2. 1. R2. v2. 3. 4. d1. d1. 3. w2. 2. R2. v 02. 2. d1 v2. 4. 3. R2. w2. v2. 4. R2. v2. 3. 2. d2. d2. 4. d2. 1 d2. Sh. Figure 2.1: Four intervals and two reservoirs replicated hydronetwork • in each time interval i, the water discharge from reservoir R1 to reservoir R2 establishes a link between the reservoirs, • the volume stored at the end of the time interval i and the volume stored at the beginning of the time interval i+1 are the same in each reservoir Rj , which establishes a link between each reservoir from the time interval i to i + 1, • the volumes stored at the beginning and at the end of the time period, vj0 and vjNt , j = 1 . . . Ne respectively, are known (they are not variables). Acceptable forecasts for electricity consumption li , i = 1 . . . Nt and for natural water inflow wji , i = 1 . . . Nt , j = 1 . . . Ne into the reservoirs of the hydrogeneration system must be available at each time interval, • the electrical power generated at each reservoir j in each time interval i depends on the initial and final volumes, vji−1 and vji respectively, and the volume of the water discharge dij . This dependence will be expressed by the notation hij (vji−1 , vji , dij ) and will be called the power hydrogeneration function of the j th reservoir over the ith time interval.. 2.1.1. The power hydrogeneration function in a reservoir. At each reservoir j the head sj depends on the headwater elevation sjw and the tailwater elevation s̄jw . The latter varies as a function of the water discharge dj ..

(49) 2.1. THE SHORT-TERM HYDROTHERMAL COORDINATION OF ELECTRICITY GENERATION PROBLEM. v i-1. vi. 23. wi. equivalent head. d. i. di. Figure 2.2: Cross-section of a reservoir For zero discharge, the downstream pool is flat and the head becomes equal to the headwater elevation, so we can write sj = sjw − s̄jw .. (2.1). The relationship between the headwater elevation sjw and the volume stored vj can be expressed approximately by the polynomial sjw = svb + svl vj + svq vj2 + svc vj3 ,. (2.2). where svb , svl , svq and svc are the basic, linear, quadratic and cubic technological coefficients respectively, which depend on each reservoir and need to be known. Also, the relationship between the tailwater elevation s̄w and the water discharge dj can be expressed in the form s̄w = sdl dj + sdq d2j , (2.3) where sdl and sdq are technological coefficients similar to before. Hence, the head sj depends on the volume stored vj and the water discharge dj , and it can be written sj (vj , dj ) = svb + svl vj + svq vj2 + svc vj3 − sdl dj − sdq d2j .. (2.4). The relationship between the variation in potential energy dEp and the variation in the water stored dvj can be expressed approximately by dEp = gsj (vj , dj )dvj , where g is the gravity. Thus, the potential energy Ep in the ith time interval, corresponding to the initial and final volumes vji−1 and vji respectively, can be obtained.

(50) 24. CHAPTER 2. THE GENERATION PROBLEM AND ITS COMPLEMENTARY CONVEX STRUCTURE. by calculating Ep =. Z. vji. vji−1. Z. dEp = g. vji. vji−1. sj (vj , dj )dvj .. (2.5). Define the equivalent head s̃j by the expression Ep = s̃j g(vji − vji−1 ).. (2.6). From the expressions (2.5) and (2.6) the equivalent head s̃j can be obtained in function of the initial volume vji−1 , the final volume vji and the volume of the water discharge dij , by evaluating the expression 1 s̃j = i vj − vji−1. Z. vji. vji−1. sj (vj , dj )dvj .. (2.7). Hence, after carrying out the necessary calculations we obtain s. i−1 2 i−1 i i s̃j = svb + s2vl (vji−1 + vji ) + vq 3 (vj − vj ) + svq vj vj + i−1 i−1 2 svc i 2 i i + 4 ((vj ) + (vj ) )(vj + vj ) − sdl dj − sdq (dij )2 .. (2.8). The main feature of this formulation is that the power hydrogeneration function is expressed approximately by a polynomial function. The power hydrogeneration function hij of the j th reservoir in the ith time interval can be obtained by considering that the energy transferred to the turbine by the water discharge is approximately equivalent to the potential energy of the weight of the water discharge dij from the equivalent head s˜j , i.e., the energy transferred is s̃j gdij . Then, the power hydrogeneration function hi of the j th reservoir in the ith time interval can be expressed j. by hij = δρij. gdij s̃j , ti. (2.9). where δ is the unit conversion coefficient, 0 ≤ ρij ≤ 1 is the efficiency coefficient, which is a concave function of the water discharge dij (during the time interval ti ) and the equivalent head s̃j . Finally, after substituting (2.8) in (2.9) we have the new expression of the power hydrogeneration function i. d s hij (vji−1 , vji , dij ) ∼ = kji tij [svb + s2vl (vji−1 + vji ) + 3vq (vji − vji−1 )2 + +svq vji−1 vji + s4vc ((vji−1 )2 + (vji )2 )(vji−1 + vji )− −sdl dij − sdq (dij )2 ],. where kji := δρij g is the efficiency and unit conversion coefficient.. (2.10).

(51) 25. 2.2. THE GENERATION PROBLEM AS A D.C. PROGRAM. 2.1.2. The Generation Problem. The objective function, which will be minimized, is the generation cost function of thermal units,   Nt X i=1. ci li −. Ne X. j=1. hij (vji−1 , vji , dij ) ,. (2.11). where li , i = 1, . . . , Nt , are the forecast for electricity consumption in the ith time interval. The linear constraints are the flow balance equations at all nodes of the network, vji − vji−1 − dij−1 + dij = wji j = 1, ..., Ne , i = 1, ..., Nt , (2.12) the nonlinear constraints are the thermal production with generation bounds, i. g≤l −. Ne X. j=1. hij (vji−1 , vji , dij ) ≤ g. i = 1, ..., Nt. (2.13). and there are positive bounds on all variables, dj ≤ dij ≤ dj. vj ≤ vji ≤ vj. j = 1, ..., Ne , i = 1, ..., Nt. (2.14a). j = 1, ..., Ne , i = 1, ..., Nt − 1.. (2.14b). Hence, the problem can be expressed by the program minimize subject to:. P Nt. . i i i=1 c l −. P. PNe. i i−1 i i e g ≤ li − N j=1 hj (vj , vj , dj ) ≤ g, vji − vji−1 − dij−1 + dij = wji ,. dj ≤ dij ≤ dj , vj ≤ vji ≤ vj ,. 2.2. . i i−1 i i j=1 hj (vj , vj , dj ). i = 1, ..., Nt , j = 1, ..., Ne , i = 1, ..., Nt j = 1, ..., Ne ,. (2.15). i = 1, ..., Nt j = 1, ..., Ne . i = 1, ..., Nt − 1. The Generation Problem as a d.c. program. A polynomial is a d.c. function on IRn because every function whose second partial derivatives are continuous on IRn is a d.c. function on IRn . Let hij (vji−1 , vji , dij ) = fji (vji−1 , vji , dij ) − gji (vji1 , vji , dij ),. (2.16).

Figure

Figure 1.1: f(x, y) :=
Figure 2.1: Four intervals and two reservoirs replicated hydronetwork
Table 2.4: Forecasts for electricity consumption (Kw) and the natural water inflow(Hm3/h) at each time interval for the reservoirs of the instances of the hydrogener-ation systems
Figure 4.1: Feasible set of the reverse convex program (4.3) and some level curvesof its objective function f(x) − t
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